Question: Solve for $y$, $ \dfrac{5}{12y} = \dfrac{8}{3y} - \dfrac{5y - 6}{9y} $
Answer: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $12y$ $3y$ and $9y$ The common denominator is $36y$ To get $36y$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{5}{12y} \times \dfrac{3}{3} = \dfrac{15}{36y} $ To get $36y$ in the denominator of the second term, multiply it by $\frac{12}{12}$ $ \dfrac{8}{3y} \times \dfrac{12}{12} = \dfrac{96}{36y} $ To get $36y$ in the denominator of the third term, multiply it by $\frac{4}{4}$ $ -\dfrac{5y - 6}{9y} \times \dfrac{4}{4} = -\dfrac{20y - 24}{36y} $ This give us: $ \dfrac{15}{36y} = \dfrac{96}{36y} - \dfrac{20y - 24}{36y} $ If we multiply both sides of the equation by $36y$ , we get: $ 15 = 96 - 20y + 24$ $ 15 = -20y + 120$ $ -105 = -20y $ $ y = \dfrac{21}{4}$